Measures of weak noncompactness and nonlinear integral equations of convolution type (Q913186)

The authors prove that the equation \(x(t)=f[t,\int^{\infty}_{0}k(t- s)x(\phi (s))ds]\) has a monotone solution \(x\in L^ 1(0,\infty)\) if suitable conditions are imposed on the functions f, k, and \(\phi\). The proof builds on measures of weak noncompactness [\textit{F. S. De Blasi}, Bull. Math. Soc. Sci. Math. R. S. R. n. Ser. 21(69), 259-262 (1977; Zbl 0365.46015)], a fixed point principle of \textit{G. Emmanuele} [Bull. Math. Soc. Sci. Math. Repub. Soc. Roum. Nouv. Ser. 25, 353-358 (1981; Zbl 0482.47027)], and certain properties of nonlinear superposition operators [\textit{P. P. Zdrejko} and the referee, J. Aust. Math. Soc. Ser. A 47, 186- 210 (1989; Zbl 0683.47045)].